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<h1><a href="https://archiveofourown.org/works/29874783">The Categorical Logic Cube</a> by <a class='authorlink' href='https://archiveofourown.org/users/Charles_Rockafellor/pseuds/Charles_Rockafellor'>Charles_Rockafellor</a></h1>

<table class="full">

<tr><td><b>Category:</b></td><td>Meta - Fandom</td></tr>

<tr><td><b>Genre:</b></td><td>Boole, Categorical logic, Deductive Reasoning, Permutational array, Socrates - Freeform, Syllogisms, Underlying pattern, Valid &amp; invalid</td></tr>

<tr><td><b>Language:</b></td><td>English</td></tr>

<tr><td><b>Status:</b></td><td>Completed</td></tr>

<tr><td><b>Published:</b></td><td>2021-03-07</td></tr>

<tr><td><b>Updated:</b></td><td>2021-03-07</td></tr>

<tr><td><b>Packaged:</b></td><td>2021-05-16 02:21:21</td></tr>

<tr><td><b>Rating:</b></td><td>General Audiences</td></tr>

<tr><td><b>Warnings:</b></td><td>No Archive Warnings Apply</td></tr>

<tr><td><b>Chapters:</b></td><td>1</td></tr>

<tr><td><b>Words:</b></td><td>2,221</td></tr>

<tr><td><b>Publisher:</b></td><td>archiveofourown.org</td></tr>

<tr><td><b>Story URL:</b></td><td>https://archiveofourown.org/works/29874783</td></tr>

<tr><td><b>Author URL:</b></td><td>https://archiveofourown.org/users/Charles_Rockafellor/pseuds/Charles_Rockafellor</td></tr>

<tr><td><b>Summary:</b></td><td><div class="userstuff">
              <p>Ever read Sherlock Holmes or Doc Savage?  We all know what makes sense (and what doesn't), and we might get it right or wrong, but can't always pin down just why.  Sometimes it's the human element, and ethics demand that we go all-out to find someone who's lost in the woods; other times, it's less of a trolley car and more a simple chain of reasoning.  This latter is much simpler to address, almost like 2+2 (mod ≥5), but often not encountered beyond truth tables in our 'teens, and even then discarded the moment that we're outside of the classroom.</p><p>What if I told you that it doesn't have to be boring and dry?  That you might be able to use it to see what Sherlock Holmes could see?</p><p>For philosophy students: you might like the illustrations - they could really make your lives a <b><i>lot</i></b> easier.</p><p>
  <b>I read and appreciate (and try to reply to) all of your comments (they are always welcome)!</b>
</p><p>𝑫𝒐𝒏'𝒕 𝒇𝒐𝒓𝒈𝒆𝒕 𝒕𝒐 𝑳𝒊𝒌𝒆, 𝑺𝒉𝒂𝒓𝒆, 𝒂𝒏𝒅 𝑺𝒖𝒃𝒔𝒄𝒓𝒊𝒃𝒆! ❤️</p><p> </p>
            </div></td></tr>

<tr><td><b>Kudos:</b></td><td>1</td></tr>

<tr><td><b>Collections:</b></td><td>Worldbuilding Meta</td></tr>

</table>

<a name="section0001"><h2>The Categorical Logic Cube</h2></a>
<div class="story"><div class="fff_chapter_notes fff_head_notes"><b>Author's Note:</b><blockquote class="userstuff">
      <p>Those of you old enough to remember him might find this work best read in the voice of Basil Rathbone.</p><p>“<i>All men are mortal</i>.”<br/>    - Socrates, posthumous (<i>attrib</i>.) 😉</p><p>And yes, this is the logic article referred to by Tails and Peach in chapter 6 of “<a href="https://archiveofourown.org/works/24354049/chapters/58727857">I am Legion</a>”.</p><p> </p><p><b>N.B.:</b>  For further meta reading, see also:</p><ul>
<li>
“<a href="https://archiveofourown.org/works/29371374">Superheroes: Powers and Principalities</a>”,
</li>
<li>“<a href="https://archiveofourown.org/works/29512119">The Hadron Octahedron</a>”,
</li>
<li>
and “<a href="https://archiveofourown.org/works/29688639">GEN 0 particles</a>”.
</li>
</ul><p> </p><hr/>
    </blockquote></div><div class="userstuff module">
    
    
<p></p><div>
<p></p><div><p><br/>
<a id="contents" name="contents"></a>
    <b>Contents:</b><br/>
[Intro]<br/>
<a href="#basics">Basics</a><br/>
<a href="#cube">The cube</a><br/>
<a href="#moods1">The four moods</a><br/>
<a href="#tetra">Tetrahedral array</a><br/>
<a href="#moods2">Tetrahedron's moods</a><br/>
<a href="#convert">Converting ladder</a><br/>
<a href="#point">Point?</a><br/>
</p></div><p> </p><p>When I took a class in categorical logic (PHI 101, UMUC, ~2005; “<i>A concise introduction to logic</i>  ”, Hurley, 9th ed.; excellent material toward the back, CD's crap), I found that the syllogisms were in triplets (which I'd forgotten since having studied it lightly in the book “<i>Logic and philosophy</i>  ” [Kahane (1978), 3rd ed. – the one with the pretty dark-cornflower blue background and pocket watch gears] in the late '80s / early '90s).  Since some were valid chains and most weren't, I sought an easy way to remember which ones were which – and “<i>Barbara celarent...</i>” was neither my style nor offered in the class.</p><p>So, out with the pen and paper for another foray into axes of freedom.  (Lemme guess: “<i><b>Another</b> foray? Dude, put down the pipe; ya have to go on a foray in the first place before you can go on another. Simple ordinality, man!</i>  ”  Well, go back and read “<a href="https://archiveofourown.org/works/29512119">An illustrated periodic table of hadrons</a>”, and you'll get it.)</p><p>The result surprised me: the valid forms present as alternating (checkerboard) coordinates in a tetrahedral corner of the phase space, with two odd exceptions in the middle two tiers: EEI and IEO.  Why they can't possibly be valid [barring null sets, if your axioms admit them] will become clear shortly, it's just that their <b>locations</b> in the cubic array give an <b>appearance</b> of being missing pieces.</p><p>Please remember that this is only <b>my own construction</b>, not an official one extant elsewhere - I mean for it to provide a visual learning tool for any whom it might help, but it's hardly to be taken as a comprehensive course.  I haven't seen it anywhere else at all, though I suppose that someone's probably done it before anyway (it's rather difficult to come up with something completely unique, and extremely unlikely if it's to do with information that's been around for millennia).</p><p>Since then, I thought about extending it to non-binary logic but didn't find the idea appealing (admittedly a little daunting, but mostly just boring).</p><p>In either case, although I've shown it to a friend or two over the years, and probably to my teacher back then, I've never sought to publish it (where would I even submit it to?), so here it is publicly for the first time in all of its unvetted glory.</p><p> </p><p>It's not my purpose here to teach a class on logic, but it's probably a good idea to present a brief primer / refresher for those who might need it before we get to the cube itself (you can <a href="https://en.wikipedia.org/wiki/Syllogism">find more on it at Wikipedia</a>, of course, with some mouthwatering Venn and Euler diagrams that might help further for visual learners).</p><p>This will be an extremely cursory cover of categorical logic.  If you're familiar with it, then you understand why.  For those who aren't already familiar with it, there's a world more material on it, and it gets deep; even skimming the details to ground you here decently would take a fair amount of your reading time - and I'm certainly not qualified to expound far on it at all.  {Contrary, subcontrary, altern, subaltern, contradictory, transposition, contrapositive, inverse, converse, obverse, negative}, Aristotle vs. Boole; the list goes on.</p><p> </p>
<h4>
<a id="basics" name="basics"></a>The basics  <b><sup><a href="#contents">Contents ▲</a></sup></b>
</h4><p>Categorical logic looks at known data as falling into one of four forms:</p>
<p></p><div><p>A = All S are P      E = No S are P<br/>
I = Some S are P O = Some S are not-P</p></div><p>It <b>doesn't</b> speak to a set in which “some” (to rhyme with “not all”) elements S are P, and therefore “some other” elements S are not-P.  If you don't know about all of the S, then you can only say what you know about some of the S: those that you have observed.  If your data indicate the existence of some that are and some that aren't, then you can assert both that some are and some aren't, but this <b>isn't</b> the same as <span class="red">{<i>some are</i> means that <i>some aren't</i> }</span>.</p><p>An example statement might be “All men are mortal”.  In this case, it's a universal affirmative “A”.</p><p>The others are universal negative “E”, particular affirmative “I”, and particular negative “O”.</p><p> </p><p>A given statement here is a quantity (all, some, or none) and a qualifier (is or isn't) attached to some term (either the predicate or the middle term [this one reappearing in a second statement]), and the same about a second term (whichever term wasn't already referred to in this statement).  The statement needn't be true (that would simply be its truth value: true, false, or unknown), nor need the terms exist in reality (though that begs a fallacy).  This first statement if the major premise.</p><p>The same is then applied to a second statement, though the two terms now will be the middle term and a subject term (not necessarily in that order).  This second statement is the minor premise.</p><p>From these, you can then draw (validly or not) a conclusion about the subject term in regard to the premise term.</p><p>When you combine two such statements into a chain in this manner, followed by a concluding statement of the same sort, you then have a syllogism:</p>
<p></p><div><p>All men are mortal.<br/>
All Greeks are men.<br/>
<b>∴</b>   All Greeks are mortal. </p></div><p>This was a case of AAA-1: each statement was “All S are P” (AAA), and the premises followed {MP, SM, SP} sequence (1).  “M” is the middle term, “S” the subject, and “P” the predicate.  The other figures are 2 {PM, SM}, 3 {MP, MS}, and 4 {PM, MS}.</p><p>There are 64 possible triplets {AAA, AAE, AAI, AAO, AEA, ..., OOO}.  These triplets are called moods.  Aristotle considered 19 of these to be valid, and Boole admitted 5 more.  These 24 are the heart of the categorical logic cube that we'll get to in a moment.  What makes the others invalid is that one can take true statements about things that exist, link these two statements together, and still draw a false conclusion – even if they <i>can</i> sometimes draw true conclusions, the fact remains that it isn't guaranteed to be the case invariably, and so <i>can</i> sometimes instead draw false conclusions.</p><p> </p><p>To return to the cases of EEI and IEO, let's look at the same terms under these moods:</p>
<p></p><div><p><b>EEI        </b><br/>
No men are mortal.<br/>
 No Greeks are men.<br/>
  <b>∴</b>   Some Greeks are mortal.</p><p><b>IEO        </b><br/>
 Some men are mortal.<br/>
 No Greeks are men. <br/>
   <b>∴</b>   Some Greeks are not mortal.</p></div><p>In the first instance, aside from the truth values of each statement, the conclusion simply doesn't relate to the major and minor premises (the first and second statements): if no Greeks are men, then men's mortality is irrelevant to the question of Greeks' mortality (hence the truth value would remain unknown).</p><p>In the second instance, we don't know if any men are immortal, but it remains asserted that no Greeks are men, and so men's mortality status is still irrelevant, hence this mood is just as invalid as the previous mood.</p><p> </p><p>I'd be remiss if I didn't at least touch on the question of fallacies.</p><p>There are many possible fallacies, but one that I've never quite grasped the limitations of is existential fallacy.  If something doesn't exist, then logic [typically, but not in all forms of logic] says that you can't really say anything much about it – yet existential fallacy isn't considered to be universally fatal to a valid deduction.</p><p>Assume for the moment that unicorns and Martians don't exist (this isn't meant as an attack on your beliefs, so if you can think of two other things that you don't believe in, then insert those here instead – if you believe in everything, then it's going to get tricky here).</p>
<p></p><div><p>  All unicorns are mortal.<br/>
   All Martians are unicorns.<br/>
   <b>∴</b>   All Martians are mortal.   </p></div><p><i>To me</i>, this second example commits an existential fallacy (twice), but it's still in the first example's form of AAA-1, a form that isn't considered to be existentially fallible – but that's <i>just me</i>.  (Mind you, I don't consider existential fallacy to invalidate a chain of logic, I consider the terms to merely not exist: any logical chain in and of itself is valid or invalid on its own, regardless of the terms' truth values or existential status; separate questions.)</p><p>In either case though, a valid deduction occurred, regardless of the material being faulty.  That's its own issue entirely.</p><p> </p>
<h4>
<a id="cube" name="cube"></a>The cube  <b><sup><a href="#contents">Contents ▲</a></sup></b>
</h4><p>Below, you'll see the categorical logic cube at last.  The formatting here is <b><span class="green highlight-pale-green"> boldface-green for always-valid moods  </span></b> and <i><span class="blue highlight-pale-yellow"> italic-blue for valid moods that risk existential fallacy  </span></i> in the array below – <b><i><span class="red highlight-pale-red"> red-fill shows the invalid moods that surprised me  </span></i></b> (i.e.: the “...<i>two odd exceptions</i>...”) when I first made the cube.  Even though the two exceptions aren't valid, they certainly are located interestingly.</p>
<p></p><div><p>
<br/>
<b><i><a href="https://i.pinimg.com/originals/0e/89/a0/0e89a06720cf5ed9e805ee2a3b2b1250.png">open image in new tab</a> to zoom</i></b></p></div><p> </p>
<h4>
<a id="moods1" name="moods1"></a>Four moods  <b><sup><a href="#contents">Contents ▲</a></sup></b>
</h4><p>Splitting the cube into a set of four cubes (one for each figure, per next illustration [the <b><i><span class="red">invalid-exception moods</span></i></b> are no longer red-filled, since there is no figure where they <i>would</i> be valid]), makes it quite clear which figure-and-mood combinations are valid, though the symmetry is then lost somewhat across this added dimension.</p>
<p></p><div><p>
<br/>
<b><i><a href="https://i.pinimg.com/originals/cc/c4/f6/ccc4f6bc5ca2b47996d27c9651aee5f3.png">open image in new tab</a> to zoom</i></b></p></div><p> </p>
<h4>
<a id="tetra" name="tetra"></a>Tetrahedral array  <b><sup><a href="#contents">Contents ▲</a></sup></b>
</h4><p>The cube can be rearranged into a tetrahedral-axial array very easily, though it loses some of its symmetry [see below].</p>
<p></p><div><p>
<br/>
<b><i><a href="https://i.pinimg.com/originals/e8/0c/cf/e80ccf71b47bb41714fe2c137386050e.png">open image in new tab</a> to zoom</i></b></p></div><p>In transforming the cubic array to a tetrahedral array [above], while the pattern that had been so prominent is now hopelessly unrecognizable, mapping each of the new array's triangular levels' cells' mostly-multiple-entries back to those of the original cubic layout (in order to verify correspondence) shows a second order symmetry acting over it.  The illustration below compares the new tetrahedron's levels (color coded on left) with the syllogisms' original locations in the cube (color coded on the right).</p>
<p></p><div><p>
<br/>
<b><i><a href="https://i.pinimg.com/originals/08/9f/1a/089f1aefe06e9b0bb1a8c87540d610d8.png">open image in new tab</a> to zoom</i></b></p></div><p> </p>
<h4>
<a id="moods2" name="moods2"></a>Tetrahedron's moods  <b><sup><a href="#contents">Contents ▲</a></sup></b>
</h4><p>Spreading out the tetrahedral array into separate figure-and-mood sets does little to pick out a pattern, other than to better show that there are only three cell coordinates in which more than one of the occupying moods is valid {2 AEO-EAO (<i>2nd tier</i> ), 4 AEO-EAO (<i>2nd tier</i> ), 2 AEE-EAE (<i>1st tier</i> )}, unless you include the <b><i><span class="red">contending IEO</span></i></b>  with at least one EIO (<i>2nd tier</i> ).</p>
<p></p><div><p>
<br/>
<b><i><a href="https://i.pinimg.com/originals/b1/16/d0/b116d028308d71f778229e7ba066cb5a.png">open image in new tab</a> to zoom</i></b></p></div><p> </p>
<h4>
<a id="convert" name="convert"></a>Converter  <b><sup><a href="#contents">Contents ▲</a></sup></b>
</h4><p>On a side note, it's also pleasantly worthwhile to see the {true, false, unknown} results of permuting statements' transforms differently (e.g.: {∀SP = T} ⊃ {nSnP = T}, {∃PS = T}, {nSP = F}, {∃SnP = F}, {∃PnS = U}, {∀PS = U}, ... <span class="highlight-pale-yellow">  [shows a DIM 13 symmetry, I think; must dig out my old notes for this] </span>), though it's much more comprehensible to use <a href="https://web.archive.org/web/20060318133125/http://www.datanation.com/fallacies/convert/view.htm">Stephen Downes's “categorical converter”</a> (which I keep thinking is called a “<i>logic ladder</i>  ”), especially if you expand it to eight ladders and add all possible paths' T/F/U conclusions, than to set them to a 32 x 32 grid (just... <i>trust me</i> on this one...).</p><p> </p>
<h4>
<a id="point" name="point"></a><i>...the point being...?</i>   <b><sup><a href="#contents">Contents ▲</a></sup></b>
</h4><p>The point of all of this is that while one certainly can write <i>around</i> something, conveying the characters' deep insights without needing to understand them oneself (and this is often a good idea to keep things flowing smoothly for the reader), nevertheless one <i>can</i> drop absolutely valid logic chains into a mystery without having to major in it (e.g.: “Cube” [the first movie], 1997).  One could even fashion a story around the logical points, thoroughly laid out in the structure of the story, without ever explicitly stating or alluding to them (not beyond the outline, perhaps).  Consider “The squares of the city” (John Brunner, 1965); the story is modelled after a chess game (<a href="https://en.wikipedia.org/wiki/World_Chess_Championship_1892">Chigorin vs. Steinitz, Havana, 1892</a>, not sure which of 23 in the set) - it's been ~35 years since I read it, so I don't remember how clear the parallels were, but one could apply the principle and follow a rigorous chain of logic applied to the story or by the MCs without ever once referring to the logic itself... or one could make the logic an Easter Egg, as in the case of LEXX's S04E18 (“The game”) use of the 21st game in the 1834 Bourdonnais vs. McDonnell championship, though in this particular example they changed the last couple of moves.  Maybe dig deeper and fashion a plot arc around warring factions of philosophy: deductive vs. inductive, null set inclusive vs. exclusive, non-binary (e.g.: middle value, fuzzy), etc. (I could easily see this last in a fic of “Avatar: the last Airbender”, as a parallel to “The Great Divide” [Book 1, ep. 11], in which there was a war between two groups... yeah, I'd better not spoil it for you - or to parallel the “Go God, go” arc of South Park [S10E12-13]).</p><p>“Good idea / bad idea”, to quote Animaniacs?  That's in the eye of the beholder.  I use ZFC and GCH to explain my screwed up story-verse (the TL;DR for that is irrelevant: the point is that you can work with anything [pi being the usual example]); Moorcock's multiverse (a strong inspiration for me) is vast, with an underlying Law/Chaos theme (and he paints a very large canvas, without ever actually laying out the deeper philosophies embroiled in their battles).  Ultimately, it's just a question of whether it appeals to <i>you</i> that matters.</p><p> </p><p> </p>
<p></p><div><p>
      <b>O ~~~ O</b>
    </p></div><p> </p></div>
  </div><div class="fff_chapter_notes fff_foot_notes"><b>Author's Note:</b><blockquote class="userstuff"><p>If you're wondering about the font colors and highlights, it's easy to do.  Here's a tutorial to get you set up quickly and easily: “<a href="https://archiveofourown.org/works/28934610">Fonts, and colors, and work skins, oh my!</a>”.</p></blockquote></div></div>
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